What happens to the shape of a parabola, as the distance between  the vertex of the curve and the focus becomes very large?  Prove your answer using algebra.

Expert Answers
tiburtius eNotes educator| Certified Educator

Let `y=ax^2+bx+c` be equation that defines our parabola.

Coordinates of vertex are:` (-b/(2a),-D/(4a))`

Coordinates of focus are:`(-b/(2a),(1-D)/4a)`

Where `D` is discriminant `D=b^2-4ac`. Note that  `x` coordinates are the same, which is to be expected because both vertex anf focus are points on the axis of symmetry.

Therefore focal length (distance between focus and vertex) is  `f=|(1-D)/(4a)-(-D)/(4a)|=1/(4|a|)`.

So if `f` is very large then `|a|` must be very small because from above formula we have` ` `|a|=1/(4f)`

So as `f` gets bigger, `|a|` becomes smaller so we have `y=ax^2+bx+c` where `a` is very small. If `a` were to become infinitely small, our equation would become `y=bx+c` which is equation of a line. 

So, what happens with parabola as focal length becomes very great?

1. Two branches of parabola become more and more distant (i.e. parabola is becoming wider and wider)

2. Ultimately as focal length becomes infinitaly great parabola tends to look like line parallel to directrix.

Graph shows parabolas for different `a` (so different `f` ).

From purple, blue, green, yellow, orange, red and black `f` is equal to `1/40,1/20,1,5/4,5/2,25`

For more on parabola see the link below.